Emma is saving money and plans on making monthly contributions into an account earning a monthly interest rate of $0.475 \%$ . If Emma would like to end up with $\$ 14,000$ after $14$ months, how much does she need to contribute to the account every month, to the nearest dollar? Use the following formula to determine your answer. $\newline$ $A=d\left(\frac{(1+i)^{n}-1}{i}\right)$ $\newline$ $A=$ the future value of the account after $n$ periods $\newline$ $d=$ the amount invested at the end of each period $\newline$ $i=$ the interest rate per period $\newline$ $n=$ the number of periods $\newline$ Answer:

Full solution

Q. Emma is saving money and plans on making monthly contributions into an account earning a monthly interest rate of

0.475 \%

. If Emma would like to end up with

\$ 14,000

after

14

months, how much does she need to contribute to the account every month, to the nearest dollar? Use the following formula to determine your answer.

\newline

A=d\left(\frac{(1+i)^{n}-1}{i}\right)

\newline

A=

the future value of the account after

n

periods

\newline

d=

the amount invested at the end of each period

\newline

i=

the interest rate per period

\newline

n=

the number of periods

\newline

Answer:

Identify Given Values: Identify the given values from the problem. $\newline$ $A$ (future value of the account) = $\$14,000$ $\newline$ $i$ (interest rate per period) = $0.475\%$ per month, which is $0.00475$ in decimal form $\newline$ $n$ (number of periods) = $14$ months $\newline$ We need to find the value of $d$ (the amount invested at the end of each period).
Convert Interest Rate: Convert the interest rate from a percentage to a decimal. $i = 0.475\% = \frac{0.475}{100} = 0.00475$
Plug Values into Formula: Plug the values into the formula to solve for $d$ . $A = d \times \left(\frac{(1 + i)^{n} - 1}{i}\right)$ $\$14,000 = d \times \left(\frac{(1 + 0.00475)^{14} - 1}{0.00475}\right)$
Calculate Value Inside Parentheses: Calculate the value inside the parentheses. $\newline$ $(1 + 0.00475)^{14} - 1$ $\newline$ = $(1.00475)^{14} - 1$ $\newline$ = $1.069678 - 1$ $\newline$ = $0.069678$
Divide by Interest Rate: Divide the result by the interest rate $i$ . $\newline$ $\frac{0.069678}{0.00475}$ $\newline$ $= 14.664842105263158$
Solve for d: Solve for d by dividing $A$ by the result from Step $5$ . $\newline$ $d = \frac{\$14,000}{14.664842105263158}$ $\newline$ $d \approx 954.545$
Round to Nearest Dollar: Round the result to the nearest dollar. $d \approx \$955$